We investigate in this paper the properties of some dilatations or contractions of a sequence of -optimal quantizers of an -valued random vector defined in the probability space with distribution . To be precise, we investigate the -quantization rate of sequences when or and . We show that for a wide family of distributions, one may always find parameters such that is -rate-optimal. For the gaussian and the exponential distributions we show the existence of a couple such that also satisfies the so-called -empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptotically -optimal. In both cases the sequence is incredibly close to -optimality. However we show (see Rem. 5.4) that this last sequence is not -optimal (e.g. when = 2, = 1) for the exponential distribution.
Mots clés : rate-optimal quantizers, empirical measure theorem, dilatation, Lloyd algorithm
@article{PS_2009__13__218_0, author = {Sagna, Abass}, title = {Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction}, journal = {ESAIM: Probability and Statistics}, pages = {218--246}, publisher = {EDP-Sciences}, volume = {13}, year = {2009}, doi = {10.1051/ps:2008008}, mrnumber = {2518547}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2008008/} }
TY - JOUR AU - Sagna, Abass TI - Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction JO - ESAIM: Probability and Statistics PY - 2009 SP - 218 EP - 246 VL - 13 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2008008/ DO - 10.1051/ps:2008008 LA - en ID - PS_2009__13__218_0 ER -
%0 Journal Article %A Sagna, Abass %T Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction %J ESAIM: Probability and Statistics %D 2009 %P 218-246 %V 13 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2008008/ %R 10.1051/ps:2008008 %G en %F PS_2009__13__218_0
Sagna, Abass. Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 218-246. doi : 10.1051/ps:2008008. http://www.numdam.org/articles/10.1051/ps:2008008/
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